Stable exponential cosmological solutions with zero variation of G in the Einstein-Gauss-Bonnet model with a Lambda-term

A D-dimensional gravitational model with a Gauss-Bonnet term and the cosmological term Lambda is considered. By assuming diagonal cosmological metrics, we find, for certain fine-tuned Lambda, a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters H>0 and h<0, corresponding to factor spaces of dimensions m>3 and l>1, respectively, with (m,l) non-equal to (6,6), (7,4), (9,3) and D = 1 + m + l. Any of these solutions describes an exponential expansion of 3-dimensional subspace with Hubble parameter H and zero variation of the effective gravitational constant G. We prove the stability of these solutions in a class of cosmological solutions with diagonal metrics.


Introduction
In this paper we consider D-dimensional gravitational model with Gauss-Bonnet term and cosmological term Λ. The so-called Gauss-Bonnet term appeared in string theory as a correction to the (Fradkin-Tseytlin) effective action [1]- [5].
We note that at present the Einstein-Gauss-Bonnet (EGB) gravitational model and its modifications, see [6]- [27] and refs. therein, are intensively studied in cosmology, e.g. for possible explanation of accelerating expansion of the Universe which follow from supernovae (type Ia) observational data [28,29,30].
Here we deal with the cosmological solutions with diagonal metrics governed by n > 3 scale factors depending upon one variable, which is the synchronous time variable. We restrict ourselves by the solutions with exponential dependence of scale factors and present a class of such solutions with two scale factors, governed by two Hubble-like parameters H > 0 and h < 0, which correspond to factor spaces of dimensions m > 3 and l > 1, respectively, with D = 1 + m + l and (m, l) = (6, 6), (7,4), (9,3). Any of these solutions describes an exponential expansion of 3d subspace with Hubble parameters H > 0 [31] and has a constant volume factor of (m − 3 + l)-dimensional internal space, which implies zero variation of the effective gravitational constant G either in Jordan or Einstein frame [32,33], see also [34,35,36] and refs. therein. These solutions satisfy the most severe restrictions on variation of G [37].
We study the stability of these solutions in a class of cosmological solutions with diagonal metrics by using results of refs. [25,26] (see also approach of ref. [23]) and show that all solutions, presented here, are stable. It should be noted that two special solutions for D = 22, 28 and Λ = 0 were found earlier in ref. [22]. In ref. [25] it was proved that these solutions are stable. Another set of six stable exponential solutions: five in dimensions D = 7, 8, 9, 13 and two -for D = 14, were considered recently in [27].

The cosmological model
The action of the model reads where g = g M N dz M ⊗ dz N is the metric defined on the manifold M , dim M = D, |g| = | det(g M N )|, Λ is the cosmological term, R[g] is scalar curvature, is the standard Gauss-Bonnet term and α 1 , α 2 are nonzero constants. We consider the manifold with the metric where B i > 0 are arbitrary constants, i = 1, . . . , n, and M 1 , . . . , M n are one-dimensional manifolds (either R or S 1 ) and n > 3. Equations of motion for the action (2.1) give us the set of polynomial equations [25] Here are, respectively, the components of two metrics on R n [17,18]. The first one is a 2-metric and the second one is a Finslerian 4-metric. For n > 3 we get a set of forth-order polynomial equations. We note that for Λ = 0 and n > 3 the set of equations (2.4) and (2.5) has an isotropic solution v 1 = . . . = v n = H only if α < 0 [17,18]. This solution was generalized in [20] to the case Λ = 0.
It was shown in [17,18] that there are no more than three different numbers among v 1 , . . . , v n when Λ = 0. This is valid also for

Solutions with constant G
In this section we present a class of solutions to the set of equations (2.4), where H is the Hubble-like parameter corresponding to an m-dimensional factor space with m > 3 and h is the Hubble-like parameter corresponding to an l-dimensional factor space, l > 1. We split the m-dimensional factor space into the product of two subspaces of dimensions 3 and m−3, respectively. The first one is identified with "our" 3d space while the second one is considered as a subspace of (m − 3 + l)-dimensional internal space.
We put for a description of an accelerated expansion of a 3-dimensional subspace (which may describe our Universe) and also put for a description of a zero variation of the effective gravitational constant G.
We remind (the reader) that the effective gravitational constant G = G ef f in the Brans-Dicke-Jordan (or simply Jordan) frame [32] (see also [33]) is proportional to the inverse volume scale factor of the internal space, see [34,36] and references therein.
Remark. Due to ansatz (3.1) "our" 3d space expands (isotropically) with Hubble parameter H and (m − 3)-dimensional part of internal space expands (isotropically) with the same Hubble parameter H too. To avoid possible puzzles with separation of these two subspaces, we consider for physical applications (in our epoch) the internal space to be compact one, i.e. we put in (2.2) M 4 = . . . = M n = S 1 . We also put the internal scale factors corresponding to present time t 0 : a j (t 3)) to be small enough in comparison with the scale factor of "our" space According to the ansatz (3.1), the m-dimensional factor space is expanding with the Hubble parameter H > 0, while the l-dimensional factor space is contracting with the Hubble-like parameter h < 0.
It was shown in [26] (for more general prescription see also [21]) that if we consider the ansatz (3.1) with two Hubble-like parameters H and h with two restrictions imposed then relations (2.4) and (2.5) may be reduced to the following set of equations . where and The substitution of relation (3.7) into (3.5) gives us It may be verified that equality P (m, l) = 0 takes place for (m, l) = (9, 3), (7, 4), (6,6).
The domains with different signs of P = P (m, l) and α are depicted in Figure 1, where we enlarged our setup by adding the case m = 3, which gives a solution with h = 0. For more general solution with m ≥ 3 and h = 0 see also [26].
Relation P (m, l) > 0, or α < 0, takes place in the following cases The domains with different signs of Λ = Λ(m, l) are depicted in Figure 2. 15 . This solution was found in [25]. For m = 15, l = 6 and α = 1 we are led to another solution from [22] with Λ = 0, H = 1 6 and h = − 1 3 . It was proved in [25] that these two solutions are stable.

Stability analysis
Here, as in [25,26], we deal with exponential solutions (2.3) with non-static volume factor, which is proportional to exp( n i=1 v i t), i.e. we put We put the following restriction For general cosmological setup with the metric we have the (mixed) set of algebraic and differential equations [17,18] where h i =β i , and it is unstable (as t → +∞) if K(v) = n k=1 v k < 0. We remind the reader that the perturbations δh i obey (in linear approximation) the following set of equations [25,26] where . . , n. It was proved in ref. [26] that the set of linear equations on perturbations (4.10), (4.11) has the following solution i = 1, . . . , n, when restrictions (4.1), (4.2) are imposed. It was shown in [26] that for the vector v from (3.1), obeying relations (3.4), the matrix L has a block-diagonal form (L ij ) = diag(L µν , L αβ ), (4.17) where L µν = G µν (2 + 4αS HH ), (4.18) L αβ = G αβ (2 + 4αS hh ) (4.19) and (4.21) The matrix (4.17) is invertible if and only if m > 1, l > 1 and

Conclusions
We have considered the D-dimensional Einstein-Gauss-Bonnet (EGB) model with the Λ-term and two constants α 1 and α 2 . By using the ansatz with diagonal cosmological metrics, we have found, for certain Λ = Λ(m, l) and α = α 2 /α 1 , a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters H > 0 and h < 0, corresponding to submanifolds of dimensions m > 3 and l > 1, respectively, with (m, l) = (6, 6), (7,4), (9,3) and D = 1 + m + l. Here m > 3 is the dimension of the expanding subspace and l > 1 is the dimension of contracting one.
Any of these solutions describes an exponential expansion of "our" 3dimensional subspace with the Hubble parameter H > 0 and anisotropic behaviour of (m − 3 + l)-dimensional internal space: expanding in (m − 3) dimensions (with Hubble-like parameter H) and contracting in l dimensions (with Hubble-like parameter h < 0). Each solution has a constant volume factor of internal space and hence it describes zero variation of the effective gravitational constant G. By using results of ref. [26] we have proved that all these solutions are stable as t → +∞.
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